Moments of the ruin time and the total amount of claims until ruin in a diffusion risk process

- Journal title : Journal of the Korean Data and Information Science Society
- Volume 27, Issue 1, 2016, pp.265-274
- Publisher : Korean Data and Information Science Society
- DOI : 10.7465/jkdi.2016.27.1.265

Title & Authors

Moments of the ruin time and the total amount of claims until ruin in a diffusion risk process

Kim, Jihoon; Ahn, Soohan;

Kim, Jihoon; Ahn, Soohan;

Abstract

In this paper, we consider a diffusion risk process, in which, its surplus process behaves like a Brownian motion in-between adjacent epochs of claims. We assume that the claims occur following a Poisson process and their sizes are independent and exponentially distributed with the same intensity. Our main goal is to derive the exact formula of the joint moment generating function of the ruin time and the total amount of aggregated claim sizes until ruin in the diffusion risk process. We also provide a method for computing the related first and second moments using the joint moment generating function and the augmented matrix exponential function.

Keywords

Augmented matrix;diffusion risk process;exponential claim;Poisson process;

Language

English

References

1.

Ahn, S. (2015). Total shift during the first passages of Markov modulated Brownian motion: Formulas driven by the minimal solution matrix of a Riccati equation. Accepted in Stochastic Models.

2.

Ahn, S., Badescu, A. L., Cheung, E. C. K. and Kim, J. (2016). Insurance risk models with marked Poisson arrivals. In preparation.

3.

Asmussen, S. (2003). Applied Probability and Queues, Second Edition, Springer Verlag.

4.

Chi, Y. (2010). Analysis of the expected discounted penalty function for a general jump-diffusion risk process and applications in finance. Insurance: Mathematics and Economics, 46, 385-396.

5.

Fung, T. C. (2004). Computation of the matrix exponential and its derivatives by scaling and squaring. International Journal for Numerical Methods in Engineering, 59, 1273-1286.

6.

Gerber, H. U. and Shiu, E. S. W. (1997). The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 21, 129-137.

7.

Irving, R. (2013). Beyond the quadratic formula, The Mathematical Association of America.

8.

Lee, E. Y. (2014). Stationary analysis of the surplus process in a risk model with investments. Journal of the Korean Data & Information Science Society, 25, 915-920.

9.

Won, H. J., Choi, S. K. and Lee, E. Y. (2013). Ruin probabilities in a risk process perturbed by diffusion with two types of claims. Journal of the Korean Data & Information Science Society, 24, 1-12.